What are the holes in the pictures called? They are usually found in sinks and tubs and their function is to prevent water from overflowing.
Although I see nothing wrong with overflows proposed by the answers and people casually call it that way, since I see in the comments that it technically refers to the whole system, I'd like to suggest the compound noun overflow holes. I did a search and found out it's actually quite common. I found some posts in Home Improvement Stack concerning these holes, too.
There is even a cover which blocks the hole called overflow hole cover for cosmetic purposes (as JimmyJames mentioned in his comment). Besides, you might want to consider overflow drain that I already mentioned under Colin Fine's answer.
(Whole credit, though, should go to Colin Fine because he was the first to post the word overflow and made me go for it)
These items are called sink overflows and are a common feature on baths and sinks. They are present to prevent the sink from overflowing with water should the plughole be blocked (by a plug for example).
For more information, see this article on overflows on a website that sells sinks and faucets.
If you look a the original patent, the inventor refers to them as overflow drain openings. A previous invention that was basically a standpipe that fit into the main drain called it an overflow drain opening as well.
- An overflow for a molded plastic sink is formed of a molded plastic body having a smooth, hollow interior forming a watertight passageway. The lower region of the body has a through opening surrounding the main drain over an opening in the drain and is sealed to the sink and the drain, the upper region of the body is sealed to the sink around an overflow opening extending through the sink and into the body. The mid region of the body is spaced from the underside of the sink.
It has a name, it’s called a porcelator.
TFD says: 1. A pipe used to remove excess water and/or to prevent flooding in certain sanitary fixtures, storage tanks, and plumbing fittings